Mathematical Model of Pulsed Immunotherapy for Superficial Bladder Cancer

Transcription

1 Mathematical Model of Plsed Immnotherapy for Sperficial Bladder Cancer Svetlana Bnimovich-Mendrazitsky, Helen Byrne and Lewi Stone Biomathematics Unit, Department of Zoology, Faclty of Life Science Tel-Aviv University, Tel-Aviv 69978, Israel Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, Nottingham, NG7 RD, UK Key words: Transitional Cell Carcinoma (TCC, Immne response, Therapy schedle, Implsive differential eqations. Abstract: We present a theoretical stdy of sperficial bladder cancer growth and its treatment via plsed immnotherapy with Bacills Calmette-Ge rin (BCG, an attenated strain of Mycobacterim bovis. BCG plsed immnotherapy is a clinically established procedre for the treatment of sperficial bladder cancer. In this paper periodic BCG instillations are modeled sing implsive differential eqations, which are stdied sing a combination of analytical and nmerical techniqes. In this way, we determine critical threshold vales of the BCG instillation dose and rate of plsing for sccessfl treatment. We also identify treatment regimes in which tmor destrction occrs bt ndesirable side effects are maintained at low levels by the immne system. Corresponding athor: address: bnimovi@post.ta.ac.il

2 . Introdction Bladder cancer is a world-wide problem, being the forth most common cancer among men and the eighth most common cancer among women (Meyer et. al.,. Its incidence is increasing in indstrialized and developed contries whereas the mortality rate remains largely nchanged (Chopin and Gattegno,. Bladder cancer is an aggressive disease with fatal conseqences. Untreated it slowly grows, first into the bladder wall and then ot into the abdomen and nearby organs, sch as the prostate, vagina, ters and rectm. The most common cancer of the bladder, Transitional Cell Carcinoma (TCC, originates from the transitional epithelim (part of the rothelim that lines the bladder cavity. TCC has been classified into two types depending on the depth of penetration and risk of progression (see Fig.: a Sperficial bladder cancer incldes low-grade papillary TCC (stages Ta and T and carcinoma in sit (CIS. Grows occrs sperficially on the inner srface of the bladder in the form of a polyp bt does not extend to the mscle. Of patients presenting with TCC, 5% are in stage Ta, % in T, and 5% CIS (Chopin and Gattegno, ; Bevers et al., ; Schenkman and Lamm,. b Invasive bladder cancer consists of high-grade TCC (stages T-T and affects abot 5% of patients presenting with TCC. Here the cancer invades the srronding mscle and tisse, casing metastatic disease. Treatment often reqires removal of the bladder (cystectomy to prevent the tmor from spreading to the lngs, liver, and bone (Nseyo and Lamm, 997.

3 Figre. Stages of bladder cancer from Meyer et al., (. Bladder cancer can be divided into sperficial and mscle invasive cancer. Sperficial bladder cancer penetrates the bladder wall no deeper than the epithelial tisse and lamina proprio Ta, T stages and CIS. The preferred treatment for bladder cancer depends on its grade at diagnosis. For sperficial bladder cancer, local srgery is applied first. Small regions of canceros tisse are removed throgh the rethra, sing a local procedre called transrethral resection (TUR. Dring TUR, the doctors also remove a small piece of polyp for microscopic examination (biopsy. In the case of malignant tisse (indicating the tmor is in 3

4 progression, the treatment is either chemotherapy or immnotherapy for eradication of any residal cancer cells. Intravesical Bacills Calmette-Gérin (BCG is a type of immnotherapy that is sed to treat sperficial bladder cancer. BCG is an attenated non-pathogenic strain of Mycobacterim bovis that was originally sed as a vaccine against tberclosis. In this treatment bacterial instillations are introdced into the bladder with a lighted tbe (catheter that is inserted throgh the rethra. After instillation, BCG accmlates near the bladder wall, and in sperficial tmor cells. On binding to the cell wall, BCG is internalized and processed by both antigen-presenting cells (APC and ninfected tmor cells. BCG antigens stimlate a strong immne response characterised by a srge in cytokine levels in infected areas and in the rine. The cytokine cascade stemming from the APCs activates a variety of cells inclding cytotoxic T cells (CTL, lymphocyte-activated killer (LAK (Schenk-Braat and Bangma, 5, natral killer cells (NK (Bohle and Branda, 3, which ltimately kill the BCG-infected tmor cells (Patard et al., 998 and eliminate the entire tmor. Crrent practice involves administering weekly intravesical instillations of BCG over a 6-week period for low-risk tmor progression and for p to 9-weeks for high-risk tmor progression (Chopin and Gattegno,. Presently, plsed BCG treatment of TCC works best for sperficial bladder cancer, althogh it is not completely effective. For example, even after treatment with BCG the probability that new tmors will develop in other areas of the bladder over the next ten years is 75%. Additionally, % of patients who receive BCG treatment experience side-effects, while another 5% develop BCG sepsis in which bacteria or other toxins are fond in the blood or other tisses of the body (Alexandroff et al., 999. Other side effects associated with BCG inclde high fever,

5 general malaise and chills, persistent bladder infection, lng infection and liver toxicity (Kim and Steinberg,. As with any biological therapy, the response to BCG is not linear and hence changing dosing schedles may nexpectedly case adverse side-effects, leading to an interrption in the corse of treatment (Zlotta et al.,. In addition, a large BCG dose may sppress the maximm indction of cytokines and adhesion molecles (Alexandroff et al., 999. After trying BCG protocols to improve the response Schenkman and Lamm ( conclded that it is difficlt to define the optimal protocol of BCG immnotherapy. Or modeling approach is similar to that of Kznetsov et al. (99, Panetta (996, Kirschner and Panetta (998, Lakmeche and Arino (, De Pillis et al. (5; 6, althogh none of these athors stdied bladder cancer or plsing immnotherapy treatment. In Bnimovich-Mendrazitsky et al. (7, the first mathematical model describing tmorimmne interactions in the bladder as a reslt of continos BCG therapy, was developed. The aim in the present paper is to adapt their model in order to provide insights into the dynamic processes that occr when BCG immnotherapy is administered as a seqence of plsed instillations.. Mathematical model The following model is based on plsed administration of BCG therapy for the treatment of sperficial TCC of the rinary bladder. It extends a model originally developed by Bnimovich-Mendrazitsky et al. (7 to describe continos administration of BCG therapy. The earlier model admits three types of eqilibria when the growth rate of the tmor cells is taken to be logistic: 5

6 I. Tmor-eqilibrim - here persistence of the tmor indicates failre of the BCG treatment; II. Tmor-free eqilibrim - here the tmor is eradicated, with only transient side effects; III. Side-effects eqilibrim - here the tmor is eradicated bt a persistent immne response is elicited. The existence and stability of these steady states depend on biologically relevant parameters and the size of the tmor at the start of treatment. For most parameter ranges one locally stable eqilibrim exists, althogh cases of coexistence can occr. Conditions for local stability and bistability are detailed in Bnimovich-Mendrazitsky et al. (7. In the worst case scenario, where the growth rate of cancer cells is taken to be exponential, the model has no stable eqilibria and nbonded tmor growth is predicted. In this paper we focs on plsed BCG treatment, rather than continos infsion, sing implsive differential eqations to model the plsed treatment (see eg., Lakshmikantham, 989, Bainov 993, Shlgin et al. 998, Stone et al.. We aim to establish criteria for sccessfl BCG treatment. 6

7 Figre. Schematic diagram of the model illstrating how tmor cell eradication by BCG may occr in sperficial bladder cancer. As stated above the model that we develop describes interactions between BCG, the immne system and tmor cells within the bladder (see Fig.. There are for key variables of interest: B(t the concentration of BCG in the bladder; E(t the concentration of activated immne-system or effector cells, that are toxic to the tmor infected cells. The tmor cells are divided into two sbpoplations: (t - those that are ninfected; T (t - those that have been infected with BCG. T i 7

8 To simplify the model we grop the immne cells (APC, CTL, NK, LAK into a single poplation of cells which we term effector cells (E. They target and destroy infected tmor cells ( T i. The latter decrease at a rate proportional to that at which they enconter effector cells. Assming random mixing, the enconter rate is proportional to the prodct p 3 ET i 3, where p is a rate constant. Similarly, tmor cells become infected with BCG at rate pbt where p is a rate coefficient. Ths, the dynamics of the two tmor poplations are given by: dti = p3eti + pbt, dt = pbt + r T. In the above eqations, we have assmed that in the absence of BCG ninfected tmor cells T ndergo exponential growth, with growth rate r. We assme that a BCG instillation of qantity b is injected into the bladder every τ time nits. By modeling the instillation as a Dirac delta-fnction b δ (t nτ we ensre that the n'th dose raises B(t by exactly b nits at time t = n τ. As stated above, free BCG binds to malignant rothelial tmor cells, infecting them at rate p (De Boer et al., 996; Drek et al., 999. BCG is lost as a reslt of both natral decay (with half life of µ and interactions with macrophages and dendritic cells (APC, and natral killer cells (NK (Wigginton and Kirschner,, which we are treating collectively as effector cells, E. We therefore assme that this loss occrs at rate EB where p is a positive rate constant. p Combining this ideas we arrive at the following evoltion eqation for B(t: db = µ. B peb p BT + bδ (t nτ n= 8

9 Factors inflencing the nmber of effector cells are: the rate at which they are recrited from the bone marrow by infected tmor cells (αti, enconters between immne cells and BCG (p EB, their natral cell death rate ( µ, and the rate at which they are absorbed by infected tmor cells ( p 5 ET i. Therefore, the dynamics of the immne system is modelled by the following differential eqation: de = + α + The ordinary differential eqations for µ. E Ti peb p5eti and E stated above constitte or mathematical model. Before contining it is helpfl to cast them in terms of dimensionless nits (see Appendix C. The following system of nondimensional ordinary differential eqations is obtained: T,T,B i db de dti dt = B( pe p T = = E( µ + p = p 3 = p ET i BT B p + p BT + r T. 5 T + αt i, n i b δ (t nτ,, ( We impose the following initial conditions: B(= b, E( = T ( =, T (, ( i > where b represents the dose of BCG administrated in each instillation. Those model parameters which can be estimated from biological data are smmarized in Appendix C and based on vales described in Bnimovich-Mendrazitsky et al. (7. 9

10 3. Nmerical simlations Eqations ( were solved nmerically sing a forth order Rnge-Ktta scheme. Fig. 3a shows a typical simlation for a treatment protocol in which BCG instillations of strength b =. are applied every τ = 7 days (see Table, Appendix C for a fll list of parameter vales. For this choice of parameter vales, the ninfected tmor cells T are eventally eliminated, and the system converges to a state that is akin to the tmor-free eqilibrim described in Bnimovich-Mendrazitsky et al. (7, with E = T i = T =. (Here we se sperscripts "" when referring to eqilibrim vales. The BCG variable B(t, on the other hand, remains periodic in time (see Figs. 3. Fig. 3 makes clear that the parameters b and τ have significant control on the dynamics of the plsed system. Fig. 3b shows that less freqent plsing ( τ =9 days rather than τ =7 days for the same strength of BCG (i.e. b=. as in Fig.3a, may case treatment to fail: the tmor contines to grow and, in the long term, the nmber of effector cells eventally falls to zero. The simlation presented in fig. 3c shows how increasing the BCG dose from b =. to b =.5 (with τ = 7 as in Fig. 3a leads to a large and persistent effector cell poplation, a sitation which we term a "side-effect" state.

11

12 Figre 3. Series of simlations showing how changing the freqency ( τ of the BCG instillations affects the system dynamics. (a When and strength (b τ = 7 days and b =. treatment is sccessfl, the tmor is eliminated and immne response eventally dies away; (b when τ = 9 days and b =. the immne response is not strong enogh to eliminate the tmor which grows nbondedly; (c when τ = 7 and b =. 5, the tmor is eliminated, bt a strong and persistent immne response is elicited. Key: ninfected tmor cells (solid heavy line, effector cells (solid line, tmor cells infected with BCG (dashed line and BCG (dash-dotted line; concentration. Parameter vales: as per Table with initial tmor size before treatment T ( =.5.. The periodic tmor-free soltion Gided by the nmerical reslts presented in fig. 3a, showing sccessfl elimination of the tmor cells, we start or analysis of Eq. ( by demonstrating the existence of a periodic tmor-free limit cycle. By observation we note that Eq. ( admits the following soltion E = T i = T =, with the BCG dynamics satisfying db = B + bδ (t nτ. (3 n= The delta fnction formlation allows s to determine the dynamics between consective plses at t = nτ and t = (n+ τ: with [ ] B n τ + nτ B db = nτ t < (n + τ, = b. If, in addition, the initial conditions are sch that B(=b, then the soltion is given by: B(t = B(e B(t = (be.. τ t t = be + be (t τ B(t = (B(nτ + be (t nτ = b( e ( e (n+ τ τ e (t nτ t <τ τ t < τ nτ t < (n + τ.

13 In the limit of large n, B(t converges to a periodic cycle denoted by B ~ (t, and the associated infection free state may be described as: be B ~ (t = τ e E = T = T = i (t nτ nτ t < (n+ τ, ( 5. Stability of the periodic tmor-free soltion. The stability of the limit cycle soltion identified above may be stdied by introdcing a small parameter ε << and linearizing system ( abot the limit cycle soltion ( so that: B(t = B ~ (t + ε B +. ε B.. E(t = ε E + ε E +... T(t = ε T + ε T i T (t = ε T i i + ε T The linearization involves inserting (5 into the ODE system (, eqating coefficients of O(ε to zero and neglecting terms of O( ε. For clarity of notation, in what follows we (5 omit the sbscripts, writing B = B = to describe the O(ε terms., E = E,Ti = Ti,T T In this way, for n τ t < (n + τ, we obtain: db de dti dt = B p = E( µ + p = p = p B ~ (tt EB ~ (t p, B ~ (tt B ~ (t + αt, + r T T. i B ~ (t, (6 3

14 Eqations (6 constitte a system of linear differential eqations with periodic coefficients. As sch they can be analysed sing Floqet theory (for details, see Appendix B. The first step involves constrcting the fndamental matrix Y( t (Zwillinger, 989: B(t B(t B3(t B(t E (t E (t E (t E (t 3 Y (t =, Ti (t Ti (t Ti3 (t Ti (t T(t T (t T3 (t T (t where B j (t,e j(t,ti j(t,t (t j = [; ] are independent soltions of the linear system ( j (6. These soltions are obtained by choosing initial conditions ( B j(,ej(,ti j(,t j( j = [;] : so that Y ( = I. Solving Eq. (6 for these initial conditions gives: (B (t, E (t, T (B (B (B 3 (t, E (t, E (t, E 3 i (t, T (t, T (t, T i3 (t, T i i (t, T (t, T We do not state the exact forms of (t, T precise form does not affect or analysis. 3 (t = (e t (t = (B (t = (B 3 (t = (B,,, (t, e (t, E 3 (t, E bp µ t+ e (t,, (t, T τ i ( e t (t, e,, p b rt e B (t, B3(t,B (t,e3(t,e (t,ti (t τ ( e t here since their Given that t=τ represents the period of the coefficients, the Floqet mltipliers are defined as the eigenvales λ of the matrix Y ( τ :. e Det(Y( τ λi = Det τ λ e B ( τ µτ + p b λ B ( τ 3 E ( τ 3 λ e rτ B ( τ E ( τ =. Ti ( τ p b λ Since Y(τ is pper trianglar, the Floqet mltipliers are: τ µτ + pb rτ pb λ = e <, λ = e, λ3 =, λ = e. (7 According to Floqet theory, the absolte vale of one of the mltipliers is eqal to nity (Glendinning, 99, and the tmor-free limit cycle associated with system ( is

15 locally stable if the absolte vales of all other Floqet mltipliers are less than nity. Stability ths reqires that: µ τ + p b rτ p b λ = e < and λ = e <. We dedce that the periodic tmor-free soltion nder BCG-plse therapy is locally stable if the BCG dose b for a given vale of τ lies within a finite range: r b µ p < τ <, (8 p where τ b is the average dose of BCG administrated per nit time. The eradication of the tmor ths depends on two key biological ratios r / p and µ / p : a r / is the ratio of the growth rate (r of the tmor to the rate at which BCG p infects the tmor cells ( p. If for a fixed vale of τ, this ratio exceeds the effective rate of BCG inflx b b, then the tmor-free eqilibrim loses stability. This lower bond on τ τ shows how competition between the BCG infection rate and the tmor growth rate affects a tmor's response to BCG-plse therapy. b µ / is the ratio of the mortality rate of effector cells ( µ to the rate at which p they are recrited ( p. If this ratio is smaller than the effective BCG inflx, b than the τ immne system will remain switched on ( E >. Eqivalently we can consider b c (T rτ = as the minimal BCG dose for which tmor p eradication occrs, and b c(e µτ = as the maximal BCG dose which does not case an p excessive and persistent immne response (E. The two thresholds imply that doses of BCG 5

16 rτ µτ that are too large or too small (b < or b > will reslt in exponential growth in the p p size of the tmor or have a detrimental effect on the immne system (see Figs. 3a & 3c. In Fig. we se Eq. (8 to show how the critical vales of b vary with the plsing time τ. We note that as τ (i.e. the time interval between installations increases, the minimm dosage b c ( T needed to exceed the threshold and eliminate the tmor also increases. We note also that the theoretical reslts are in good agreement with nmerical simlations of Eq. (. These simlations were obtained sing initial conditions close to the tmor-free state, a point that is discssed frther below. 6. Interpreting the threshold criteria The threshold conditions (8 derived above give little insight into their origins. The analysis presented below provides a more intitive interpretation of the stability conditions. Consider first the linearized tmor dynamics in Eq. (6 dt = p B ~ (tt + r T, where B ~ (t is specified in Eq. (. Between any two BCG plses n τ t (n + τ we have: dt (t nτ be = T (r p, n τ < t < (n + τ. (9 τ e If we introdce R = rτ p b, then it is possible to show that: T ( τ = T T (τ = T ( τe... (e = T (e T ((n + τ = F [T (nτ ], R, R R, where F [T (nτ ] = T (nτ e, R ( The stroboscopic map F(T has a fixed point T = F(T, which is stable if: = 6

17 ' F (T R = e <. Ths, the criterion for tmor eradication becomes R< or eqivalently: rτ b > b c(t =. ( p Figre. Diagram showings how the otcome following plsed BCG therapy depends on the dose ( b and period of plsing (τ. Theoretical predictions are represented by dashed lines and simlations by solid lines. Crve : theoretically-determined crve showing how Crve : nmerically-determined crve showing how Crve 3: nmerically-determined crve showing how Crve : theoretically determined crve showing how b varies with τ (see Eq. (, c(t b varies withτ, c(t b varies withτ, c(e b varies with τ (see Eq. (3. c (E Parameter vales as per Table, with T ( =.. 7

18 Similarly, the linearized dynamics of the bacterial-immne system, specified by Eq. (6 may be approximated as: de = E( µ + p B ~ (t. In obtaining the above eqation we have assmed that α << and the nmber of infectemor cells remains close to the eqilibrim vale (, we dedce that: i =. Sbstitting with B ~ (t T from Eq. (t nτ de be = E ( µ + p, nτ t < (n + τ. τ e If we introdce P= p b µτ, then it is possible to show that E (nτ satisfies the stroboscopic map: E((n + τ = F[E(nτ ], where F[E(nτ ] = E (nτ e P. The map F(E has a fixed point E = F(E =, which is stable if: df (E ( µτ+ p b = e de <. ( This will be the case if: µτ b < bc(e =, (3 p where is the critical BCG therapy level. Note that if b c(e b < b c(e, then the immne system will be reglated after treatment and E will eventally redce to its eqilibrim state zero. (Ths the plot of E in Fig. 3a will eventally converge to zero. If b > then the effector cell poplation ( E > increases in size and persists at long times. The behavior of the immne system in this case is depicted in Fig. 3c. The shaded triangle in Fig. highlights those vales of BCG dose and freqency from condition (8, yielding a state that is tmor-free ( b c (E T = and withot effector cells 8

19 ( E =. Doses of BCG otside this region reslt in either exponential growth of the tmor or have a detrimental effect on the immne system. 7. Effect of initial tmor size The Floqet analysis presented in section 5 yields information abot local stability and is based on the assmption that initially the system is close to the tmor- free eqilibrim, with T (. However, if the initial tmor size is large, the threshold predictions may be affected significantly. Nmerical simlations sggest that if T ( is sfficiently large then the tmor-free limit cycle may never be reached, even thogh it is locally stable. Apart from the periodic tmor-free limit cycle discssed above, it is possible to identify a soltion for which B(t, while T (t contines to grow exponentially. In fact, there is a separatrix, which demarcates the basin of attraction of the trivial tmor-free soltion and the soltion for which the tmor grows nbondedly. If T ( is sfficiently small sch that the initial condition lies below the separatrix then the system is attracted to the tmor-free eqilibrim (see Fig. 3a. Eqally, if T ( is initially large enogh sch that the system is above the separatrix, then the tmor grows ncontrollably (Fig. 3b. We investigate this dependence on initial conditions as follows. First we sppose that initially the immne system is nstimlated (E(=, so that there are no infected tmor cells( T i ( = and that the tmor is large so that T ( >>. Now we seek to identify the critical parameter vale of b, sch that for b < b the tmor grows nbondedly ( T as t, and for b > b the tmor regresses ( T, as t. This critical vale of b delineates the switch between tmor extinction and exponential growth. 9

20 We make analytical progress by introdcing the small parameter δ << and T (t assming that treatment starts when T = >> δ and E = δ E(t <<. Inserting these rescalings into the dimensionless eqations (, we dedce that B, E, T and T i satisfy: db p T = B( + δ + δ p d E αti = E( µ + pb p5t i + δ dt p i = δ p3ti E+ T B, δ d T = (r p BT. E,, ( Note that the ODEs for B, E and T are singlar in the limit asδ, indicating that there i is a short timescale over which these variables change. We determine this behavior by rescaling time so that t = t /δ in which case eqations ( became: db = B( δ + p T d T = δ (r p B T + δ d E = αti + δ E( µ + p B p T 5 dt i = δ p 3Ti E + p T B, We then seek a power series expansion of the form:. p E, i, (5

21 B(t = B (t + δb (t +... E(t = E (t + δe(t +... T(t = T (t + δt i i i (t +... T (t = T(t + δt(t +... Or main aim is to determine vale of b, which separates parameter space into regions in which tmor growth is exponential and regions in which the tmor decays. Since. d T = O( δ, we dedce that T (t = T is constant for all t. Eqating to zero terms of O(δ we dedce that T (t solves: d T = (r p B T. (6 Solving the above ODE reqires knowledge of. Trning now, then, to the ODE for B, B we find by eqating to zero coefficients of O( that B solves: db = p T B, nτ t < (n + τ. Hence + B (t = B (nτ exp[ pt (t nτ ], (7 where the symbol "+" designates that we are referring to the concentration of bacteria Bo instantaneosly after a plse is administered. We dedce frther that when the (n+'th plse is administered (at time t = (n + τ : B [(n + τ ] = B (nτ exp( p T τ b. B We look for a periodic soltion of and ths set B[(n + τ ] B(nτ, which gives: + = + + B (nτ + = b. (8 p T τ e

22 Retrning now to eqation (6, on sbstitting from (7 for B (t, we arrive at: d T = + [ r p B (nτ exp( p T (t nτ ] T, and ths T (t = T (nτ + + r T (t nτ B (nτ + ( exp( p T (t nτ. Recall that this expression is only valid for nτ < t < (n + τ. Consider T ((n + τ which is the tmor size jst before the next plse occrs at t = (n + τ. T ((n + τ = T (nτ + + rt + τ B (nτ ( exp( p T τ. (9 Now recall that we seek the bifrcation point which separates the regions of exponential tmor growth from exponential tmor clearance. That is, we are concerned with whether the ratio of T T ((n + τ (nτ + is greater than nity (growth or less than nity (clearance. At the transition point, we have: Combining Eqs.(9 & ( gives: T = + ((n + τ T (nτ. ( rt τ = B (nτ + ( exp( p T τ. ( Inserting (8 into (, we dedce that if is periodic with period τ then rt τ = b. B Recalling that T (t = T and reverting to the original, dimensional variables, we dedce that at the transition point: Ths if b = T ( rτ.

23 = b > b T ( rτ, ( then the tmor will be destroyed ( T T as t and conversely if b < b then the tmor will grow. = From ( it is clear that the minimm amont of BCG reqired for sccessfl treatment is directly proportional to the initial tmor size T ( ; for large vales of T ( this becomes important in setting the treatment protocol. This prediction is in good agreement with clinical reslts, which show that the initial size of the tmor is a key prognostic indicator of treatment sccess. In particlar polyps of size greater than 3 cm, or the presence of mltiple polyps (e.g., mltifocal presentation of bladder cancer, are associated with poor prognosis (Meyer et al.,. Fig. 5 shows that or theoretical prediction of the bifrcation point (see Eq. ( is in good agreement with that obtained by solving the fll system of eqations and determining b nmerically. This is tre for large realistic vales of initial tmor size T ( ranging from T ( =[, 6] dimensionless nits. For example, a tmor of radis r=5mm and of depth h=3 cells, will contain 6 T ( = π r h π 5 3 = tmor cells. The calclation is 3 6 based on mm ~ cells and the length of the cell being approximately µ m (see Bnimovich-Mendrazitsky et al., 7. In dimensionless nits (scaling rles Eq. (A this corresponds to ' T ( T ( = = 6 and is ths considered a relatively large tmor, certainly 6 at the pper end of the size scale. Using Eq. ( with growth rate r =.68 andτ =. 7, we find that b =.9 (Fig. 5 (or, b =.9x viable bacterial cells or colony-forming nits, c.f..'s. It is interesting to compare this to the dose sed in medical practice. Now, in 6 3

24 clinical practice typical doses for weekly BCG instillation (Cheng et al., vary in the range to [. 8 8 to 6. ] c.f..'s. Ths, the minimm amont of BCG reqired to eliminate a large tmor is significantly less than the typical vales sed in practice. Figre 5. Diagram showing how depends on its size ( T b the predicted dose of BCG needed to eliminate a tmor at the start of treatment: Crve (star line: threshold crve fond via simlations. In the region below the threshold crve exponential tmor growth occrs, while above it lead to the tmor elimination is predicted. For example, for T ( =35(dashed line, an instillation dose with b. 7 will lead to a tmor-free eqilibrim. Crve (solid line: theoretical estimate of threshold crve (see Eq. (. There is good agreement between the theoretical predictions and those obtained by solving the fll model nmerically. Parameter vales: as per Table, with τ =7.

25 7. Estimating BCG administration Using above calclations it is possible to estimate the nmber of BCG instillations reqired to eradicate a tmor. Given the deterministic natre of the model and since it is not individal based, we mst specify a small threshold tmor size below which we can consider the tmor to have been eliminated. We set the threshold arbitrarily at T = _ crit 6 (which corresponds to approximately a tmor mass containing cell: recall that = corresponds to a tmor containing cells; see Appendix A. T By virte of Eq. (, it is possible to calclate the nmber N of BCG instillations, and/or the dose b, reqired to reach the eradication threshold. First, rearranging Eq. (, we obtain: 6 T (n n(rτ p b τ = T (e. Setting ((n + τ = T =, we can find the critical dose bc: T _ crit 6 b c 6 = ( rτ log( / T ( / n. (3 p In Fig. 6 we se Eq.(3 to show how b c, the minimm dosage reqired to eliminate the tmor, varies with n the nmber of instillations. Ths for n=6 (i.e., 6 weekly instillations, we estimate b c =8.3and increasing the treatment schedle from 6 to weeks leads to a redction in the minimm dosage needed to achieve a cre by % (i.e., a redction to b c =.9. 5

26 Figre 6. The relationship obtained from Eq. (3 showing the manner in which b c, the minimm BCG treatment dose reqired to eliminate a tmor of fixed size, varies with n, the nmber of weekly instillations administrated. Parameter vales: as per Table, with T ( =. andτ =7. 8. Smmary In this paper, the complex biological processes of tmor, immne system, and BCG interactions have been modelled sing a system of for differential eqations and sed to investigate the system's response to plsed BCG immnotherapy. Depending on the dose (b and the interval between treatments (τ, one of three different otcomes were obtained:. The tmor-free state, for which the tmor is completely eradicated and the BCG dynamics evolve to a limit cycle of periodτ, where τ is the period of the instillations. The tmor-free state is locally stable provided the dose administrated lies within a finite range: 6

27 r b µ p < τ < ( p. The tmor state in which the tmor-free eqilibrim is nstable and nbonded tmor growth is predicted. This case arises when the lower bond in Eq.( is violated which means that insfficient BCG is being delivered to the tmor (see Fig. 3b. 3. The side-effect state in which the tmor is eradicated ( T bt the immne system is destabilized and the effector cells are in a state of growth. This case occrs when the pper bond in Eq. ( is violated (see Fig. 3c. The above reslts highlight the importance of the ratios r / p and µ / p in the sccess of BCG immnotherapy. The former implies that it is the relative ratio between the natral growth (r of bladder tmor cells and the BCG infection rate ( that controls whether or not therapy will scceed: whichever process has the advantage, dictates whether the tmor is eradicated. Similarly, there is a competition between the rate of recritment of effector cells ( p and their natral mortality rate ( µ. If the rate of recritment is high then for a given treatment protocol. The right side ineqality in Eq. ( will be violated and effector cells will accmlate. It is interesting to compare the stability criterion for the tmor-free state when plsed BCG is administrated with that obtained for or earlier model of continos BCG treatment (Bnimovich-Mendrazitsky et al., 7. Both models are identical except for the terms describing BCG entry into the bladder. In the continos treatment BCG enters at a constant rate b per nit time, while for the plsed treatment a qantity b of BCG enters the bladder as a spike every τ time nits. While for the continos model the tmor-free state is shown to be stable if: r < b < µ, in the plsed model the condition is jst p p p p 7

28 r b p µ p < τ <. Ths the two models are eqivalent if we view the plsing b p τ p as the average BCG per nit time. A key difference between the models is the dynamics of the tmor-free state: with continos treatment the tmor-free state is an eqilibrim, whereas with plsed treatment the BCG dynamics evolve to a limit cycle. We have also shown that the response to BCG treatment depends on the initial size of the tmor and may be sed to determine whether BCG treatment shold be given. These model predictions confirm, Nseyo and Lamm's (997 conclsion that the state of a patient s tmor here is an important determinant factor in deciding whether to opt for intravesical therapy. Indeed an ltimate, bt albeit still distant, application for the model is to aid in defining the appropriate rate of BCG instillation in a particlar clinical sitation. To achieve this goal for individal bladder cancer patients, certain clinical parameters need to be measred. Many of or model parameters were estimated from the biological and clinical literatre. However, parameters sch as the infection rate of the tmor cells with BCG (, and the rate of stimlation of effector cells by infected tmor cells ( p, have p not yet been measred, even thogh they are extremely important in predicting the otcome of therapy. In ftre work, we aim to develop a model that distingishes between varios types of immne cells and differences in their behavior copled with sitable experimental data and parameter estimates. We anticipate that this will generate more accrate predictions for optimizing BCG treatments and will allow s to determine conditions nder which the distinction is important. We will also investigate the effect on the system dynamics of introdcing a time delay to describe the time if takes for effector cells to be recrited from 8

29 the bone marrow. This delay cold profondly inflence system dynamics and give a more realistic characterization of the immne response. As has been fond in other applications (Mackey & Glass, 977; Mrray, 993; Tao & Go, 5, the delay cold profondly inflence system dynamics and give a more realistic characterization of the immne response. In conclsion, effective treatment of bladder cancer remains a challenge despite significant improvements in preventing disease progression and improving srvival. Intravesical BCG immnotherapy has been sed to manage sperficial transitional cell carcinoma (TCC of the rinary bladder, to treat existing or residal tmors, to prevent tmor recrrence, and/or disease progression, and to extend patient srvival. We hope that frther extensions of the model presented in this paper will form the basis for ftre mathematical stdies that will inform clinical practice. Acknowledgments We thank Professors Nir Ben-Tal and Moshe Aronson, Drs Eliezer Shochat and David Bnimovich for fritfl and stimlating discssions. This stdy was spported by a fellowship from the Edmond J. Safra Bioinformatics program at Tel-Aviv University. We grateflly acknowledge the financial assistance received from the British Concil throgh the Research Exchange Programme. The RXP award spported several visits to the UK and was crcial to the research collaboration which reslted in the writing of this paper. 9

30 References. Alexandroff, A.B., Jackson, A.M., O Donnell, M.A., James, K., 999. BCG immnotherapy of bladder cancer: years on. Lancet, 353, Archleta, J., Mllens, P., Primm, T. P.,. The relationship of temperatre to desiccation and starvation tolerance of the Mycobacterim avim complex. Arch. Microbiol., 78, Bainov, D., (993. Implsive Differential Eqations, Longman.. Bevers, R.F.M., Krth, K.H., Schamhart, D.H.J.,. Role of rothelial cells in BCG immnotherapy for sperficial bladder cancer. BJC, 9, Bohle, A., Branda, S., 3. Immne mechanisms in bacills Calmette Gerin immnotherapy for sperficial bladder cancer. J Urol., 7, Bnimovich-Mendrazitsky, S., Shochat, E., and Stone, L. 7. Mathematical Model of BCG Immnotherapy in Sperficial Bladder Cancer, Bll Math Biol, Online: 7. Cheng, C.W., Ng, M.T., Chan, S.Y., Sn, W.H.,. Low dose BCG as adjvant therapy for sperficial bladder cancer and literatre review. Anz Jornal of Srgery, 7 (7, Chopin, D. & Gattegno, B.,. Sperficial bladder tmors. Er.Urol.,, De Boer, E.C., Bevers, R.F., Krth, K.H., Schamhart, D.H., 996. Doble florescent flow cytometric assessment of bacterial internalization and binding by epithelial cells. Cytometry, 5,

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33 3. Schenk-Braat, E.A.M., Bangma C.H., 5. Immnotherapy for sperficial bladder cancer. Cancer Immnology Immnotherapy, 5 (5, Shlgin B., Stone L. and Agr Z., 998. Plse vaccination strategy in the SIR epidemic model, Blletin of Mathematical Biology 6, Stone L., Shlgin B., Agr Z.,. Theoretical examination of the plse vaccination policy in the SIR epidemic model, Mathematical and compter modeling, 3 (-5: Schenkman, E. and Lamm, D.L.,. Sperficial bladder cancer therapy. TheScientificWorldJ, Vol., pp Shochat, E., Hart, D., Agr, Z., 999. Using compter simlations for evalating the efficacy of breast cancer chemotherapy protocols, Mathematical Models and Methods in Applied Sciences, 9(, Swanson, K.R., Bridge, C., Mrray, J.D., Alvord, E.C., 3. Virtal and real brain tmors:sing mathematical modeling to qantify glioma growth and invasion. J Nerol Sci., 6,. 36. Zlotta A.R., van Vooren J.P., Hygen K. et al.,. What is the optimal regimen for BCG intravesical therapy? Er Urol; 37: Zwillinger, D., 989. Handbook of Differential Eqations, New York: Academic Press. 38. Wigginton, J., Kirschner, D.,. A model to predict cell-mediated immne reglatory mechanisms dring hman infection with Mycobacterim tberclosis. J. Immnol. 66,

34 Appendix A: Scaling of parameters This system may be rewritten in dimensionless form, sing the following scaling: B ' = B B, E ' = E E ' i, T ' r µ ' r =, µ =, p µ µ Ti = T p = E µ T = T ' b ' α = µ t, b =, β = βt, α =, µ B ' ', T, t i µ, p ' p = T µ, p ' 3 p3 = E µ, p ' p = B µ, p p5 = T. (A µ i In practice, there is a trade-off between redcing parameters and retaining parameters that have operational meaning. For these reasons we choose the scaling ' 5 B 6 = E = Ti = T = cells (A (similar to Kznetzov et al., 99. Dropping the prime notation for convenience, we obtain the dimensionless form of the model eqations as stated in the main text (see Eq. (. Appendix B: Floqet theory Here we apply Floqet theory for stdying the stability of Eq. (6 which describes a linear system with periodic coefficients. A review of the theory may be fond in Hartman (96, Iooss and Joseph (98, Gckenheimer and Holmes (983, and Glendinning (99. Consider the linear system of n copled differential eqations: d y = P(t y, (B where P(t = P(t + τ are periodic coefficients. Hartman (96 has shown that there is a fndamental soltion matrix of the system that can be written in the form: Y (t = Z(t exp(r t with Z(t = Z(t + τ, where Y, Z and R are n x n matrices and R is a constant matrix. In particlar, choosing Y ( = Z( = I gives: 3

35 Y( τ = Z( τ exp(rτ = Z( exp(rτ = exp(rτ. n The latter relation implies that Y(nτ = Y( τ, which in trn implies that the asymptotic behavior of y (t depends only on the eigenvales of Y( τ = exp(rτ. In fact y(nτ tends to zero provided all the eigenvales of Y( τ lie inside the nit circle (Glendinning 99. These eigenvales are referred to as Floqet mltipliers. In practice the fndamental matrix Y can be constrcted by finding n independent soltions: ( y (t, y (t,...,y n (t of system (B. 35

36 Appendix C: Parameter estimation Parameter vales were estimated in Bnimovich-Mendrazitsky et al. (7 and are smmarized in the following Table. below. Param Physical Interpretation eter (nits µ BCG half life Dimensional estimate (Dimensionless estimate Reference.(. Archleta et al., [ days ] µ Effector cells mortality rate.(. Kznetsov et al., 99 days p p p 3 p [ ] The rate of BCG killed by APC cells ][days [ ] Infection rate of tmor cells by BCG [ cells ][days ] Rate of destrction of infected tmor cells by effector cells [ cells ][days ] Immne response activation rate cells ][days [ ] (.5 Wigginton & Kirschner, 6 (.85 Not fond 7 (. Kznetsov et al., 99 7 (. Not fond 8 p Rate of E deactivation after binding.3 (.3 Kznetsov et al., 99 5 with infected tmor cells; [ cells ][days ] α Rate of E stimlation de to.5(.5 Wigginton & infected tmor cells Kirschner, [ days ] r Tmor growth rate.33 (.33 Shochat et al., 999; [ days ]. (. Swanson et al., 3;.68 (.68 In all simlations. / β Tmor carrying capacity 8 7 β = [. ] [ cells ] Lämmle et al., ; ( = = [ 5,76] De Pillis et al., 6. β [.3,.] Table. List of all parameters. Note that dimensionless estimates were obtained from sorce vales sing the transformations stated in Appendix A. 36